OFFSET
0,4
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
a(n) = n^4/12 + n^3/6 - 7*n^2/12 - 2*n/3 + 1 = (n-1)*(n-2)*(n+3)*(n+2)/12.
From Colin Barker, Feb 05 2018: (Start)
G.f.: (1 - 5*x + 10*x^2 - 5*x^3 + x^4) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5. (End)
a(n) = A033275(n+2) for n > 1. - Georg Fischer, Oct 09 2018
From Amiram Eldar, Jan 12 2021: (Start)
Sum_{n>=3} 1/a(n) = 43/150.
Sum_{n>=3} (-1)^(n+1)/a(n) = 16*log(2)/5 - 154/75. (End)
E.g.f.: exp(x)*(12 - 12*x + 6*x^2 + 8*x^3 + x^4)/12. - Stefano Spezia, Feb 21 2024
MAPLE
seq(n^4/12+n^3/6-7*n^2/12-2*n/3+1, n=0..10^3); # Muniru A Asiru, Feb 04 2018
MATHEMATICA
Rest@ CoefficientList[Series[(1 - 5 x + 10 x^2 - 5 x^3 + x^4)/(1 - x)^5, {x, 0, 41}], x] (* Michael De Vlieger, Feb 10 2018 *)
f[n_] := n^4/12 + n^3/6 - 7*n^2/12 - 2*n/3 + 1; Array[f, 40, 0] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 0, 0, 5, 21}, 40] (* Robert G. Wilson v, Mar 12 2018 *)
PROG
(Magma) [n^4/12 + n^3/6 - 7*n^2/12 - 2*n/3 + 1: n in [0..40]];
(GAP) List([0..10^3], n->n^4/12+n^3/6-7*n^2/12-2*n/3+1); # Muniru A Asiru, Feb 04 2018
(PARI) Vec((1 - 5*x + 10*x^2 - 5*x^3 + x^4) / (1 - x)^5 + O(x^50)) \\ Colin Barker, Feb 05 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Juri-Stepan Gerasimov, Feb 03 2018
EXTENSIONS
Edited by Wolfdieter Lang, Apr 06 2018
STATUS
approved